Tensor structure on perverse Nori motives

Abstract

Let $k$ be a field of characteristic $0$ endowed with a complex embedding $\sigma: k \hookrightarrow \mathbb{C}$. In this paper we complete the construction of the six functor formalism on perverse Nori motives over quasi-projective $k$-varieties, initiated by Ivorra--Morel. Our main contribution is the construction of a closed monoidal structure on the derived categories of perverse Nori motives, compatibly with the analogous structure on the underlying constructible derived categories. This is based on an alternative presentation of perverse Nori motives, related to the conjectural motivic perverse $t$-structure on Voevodsky motivic sheaves. As a consequence, we obtain well-behaved Tannakian categories of motivic local systems over smooth, geometrically connected $k$-varieties. Exploiting the relation with Voevodsky motivic sheaves in its full strength, we are able to define Chern classes in the setting of perverse Nori motives, which leads to a motivic version of the relative Hard Lefschetz Theorem.